VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved. A RECURSIVE PROCEDURE FOR COMMENSAL- HOST ECOLOGICAL MODEL WITH REPLENISHMENT RATE FOR BOTH THE SPECIES - A NUMERICAL APPROACH N. Phani Kumar C.Srinivasa kumar and N.Ch. Pattabhi Ramacharyulu 3 Faculty of Mathematics, Malla Reddy Engineering College, Hyderabad, A.P. India Principal, Gopal Reddy College of Engineering & Technology, A.P, India, 3 Mathematics (Retd) National Institute of Technology, Warangal, A.P. India E-Mail: phanikumar_nandanavanam@yahoo.com ABSTRACT In this paper a four stage recursive procedure is adopted to give an approximate solution to the mathematical model equations of a commensal-host ecological model with replenishment rate for both species. Numerical examples are discussed to explain the trajectories of the solutions and the results are illustrated. Keywords: commensal, host, replenishment rate, recursive procedure. INTRODUCTION The analytical solutions for the first order nonlinear coupled differential equations are not possible directly because of interactability of non-linear terms. Techniques like perturbation to linearizing the non-linear coupled differential equations to give qualitative nature of solutions in treatises of Kapoor [3], Meyer [5] Cushing [], Phanikumar N, Pattabhi Ramacharyulu N.Ch [6] et al. In the present investigation we present a four stage recursive procedure to give an approximate solution for a first order non linear coupled differential equations and same is applied to solve the commensal-host ecological model with replenishment rate for both the species. A similar algorithm was earlier developed by Langlosis and Rivlin [4] to obtain approximate solution for the flows of slightly visco- elastic fluids. Srinivas N.C. [7] and Bhaskara Rama Sarma [] adopted this algorithm for a general two species eco system. This paper is divided into two sections. In section - I, the general four stage recursive procedure is presented for solving a first order non - linear differential equations. In section - II the recursive procedure is applied to a commensal- host ecological model with replenishment rate for both species. Some numerical illustrations have been carried out for a wide range of the system characterizing parameters a, a, a, a, a for this matlab has been used and the results are illustrated. Section-I The general recursive procedure with four stage approximation for first order non-linear coupled differential equations: Consider the system of non-linear coupled differential equations is of the form Nt () = A N (t) + f ( N, t) + h (t) () Where Nt () = dn with initial conditions N () = N () A N (t) : Linear dependence term of the system. f ( N, t) : Non Linear dependence term of the system. h (t) : Replenishment / renewal rates of the system. The recursive procedure for solving the system () is presented in the following four stages. Stage -: Consider the linear system of () Nt () = A N (t) (3) With the initial condition N () = N The above system is obtained by suppressing the nonlinear part and in the absence of replenishment te on the right hard side of () Let the solution of (3) together with () be N () () t = N O e AT (4) Stage-: Compute the replenishment rate that could sustain the above solution of the system () ht () = Nt ()- A N () () t - f ( From (3), equation (5) becomes N () () t in the non-linear part () N, t ) (5) () ht () = - f ( N, t ) (6) Stage-3: Consider the system Nt () = A () Nt - f ( () N, t ) (7) 44
VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved. With the homogeneous initial conditions N () = (8) ( ) Let N () t be the solution of the system (7) together with (8) Stage-4: The approximate solution of non-linear system () with () is given by Nt () = () N () t - ( ) N () t (9) () Where ( ) N () t and N () t are the solutions obtained by stage- and stage-. Section- II In this section we discussed the approximate solution of commensal - host ecological model with replenishment rate for both the species S and S. Here both the species S and S with limited resources are considered. Notation Adopted: N i : Population of Species S i, =, at time t a i : Natural growth rates of species S i, i =, a ii : Rates of decrease of species S i, i =, due to limited resources a : inhibition coefficient h i : Replenishment rates of S i, I =, All these Co-efficients a, a, a, a, h, h > Consider the system dn dn = an a N + a NN + h = an a N + h () With initial conditions N i () = N i ; i =, () Now the recursive procedure explained in section - I is carried out in the foregoing analysis of the system () Stage-: Taking the system () with linear terms along with initial conditions () we get dn dn = a N = a N () and N () = N ; N () = N (3) Now the solution of () with (3) is given by N () = N e a t (4) N () = N e a t (5) Stage-: By using (4) and (5) the replenishment rates h and h of () are calculated as h (t) = a N e a t --a N N e (a + a ) t h (t) = a N e a t (6) Stage-3: Using (6) along with homogeneous initial conditions we setup the linear system dn dn = a N + h (t) = a N + h (t) (7) with initial conditions N () = ; N () = (8) The solution of the system (7) along with initial conditions (8) are N () = N e a t a N a a N ( e ) + ( e ) a at at an a t ( e ) a N () = N e a t (9) Stage-4: An approximate solution for the system is given by N = N () - N () N = N () N () then () N = N e a t a N a N ( e ) + ( e a a at at a N a ( a t N = N e a t e ) ) () A numerical solution of the basic non-linear coupled differential equations: The variation of N and N verses time t : The variation of N and N verses t in the internal (, ) are computed numerically by employing recursive procedure with four stage approximation. Technique for a wide range of the system characterizing parameters a, a ; a, a ; a, a for this matlab has been used and results are illustrated. 45
VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved. Table N = N =.5 a = a = a =.5 a =.5 a = a :.5,,.5, N = N =.5 a = a = a = a = a = a :.5,,.5, 3 N = N =.5 a = a = a =.5 a =.5 a = a :.5,,.5, 4 N = N =.5 a = a = a =.5 a =.5 a = a :.5,,.5, 5 N = N =.5 a =.5 a = a =.5 a =.5 a = a :.5,,.5, 6 N = N =.5 a = a =.5 a =.5 a =.5 a = a :.5,,.5, 7 N = N =.5 a = a = a = a = a = a :.5,,.5, 8 N = N =.5 a =.5 a = a =.5 a =.5 a = a :.5,,.5,..8 --------------->N,N.6.4. -. -.4...3.4.5.6.7.8.9 Figure-. N =, N =.5, a =, a =, a =.5 a = Variation of N and N verses time t..5 --------------->N,N.5 -.5 -...3.4.5.6.7.8.9 46
VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved..5 Figure-. N=, N =.5, a=, a=, a= a= Variation of N and N verses time t..5 --------------->N,N -.5 - -.5 - -.5-3...3.4.5.6.7.8.9.5 Figure-3. N =, N =.5, a =, a =, a =.5 a = Variation of N and N verses time t. --------------->N,N.5 -.5 - -.5...3.4.5.6.7.8.9 Figure-4. N =, N =.5, a =.5, a =, a =.5 a = Variation of N and N verses time t. 47
VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved..4. --------------->N,N.8.6.4. -. -.4...3.4.5.6.7.8.9 Figure-5. N =, N =.5, a =, a =, a =.5 a = Variation of N and N verses time t..8.6 --------------->N,N.4. -. -.4...3.4.5.6.7.8.9 Figure-6. N =, N =.5, a =, a =.5, a =.5 a = Variation of N and N verses time t. 48
VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved. a =.5 a = a =.5 a = --------------->N,N - -4-6 -8 -...3.4.5.6.7.8.9 Figure-7. N =, N =.5, a =, a =, a = a = Variation of N and N verses time t. --------------->N,N - - a =.5 a = a =.5 a = -3-4...3.4.5.6.7.8.9 Figure-8. N =, N =.5, a =.5, a =, a =.5 a = Variation of N and N verses time t. CONCLUSIONS Following conclusions are derived from the graphs (Figure- to Figure-8) a) If growth rate of host increases the N curves are falling with increasing steepness. b) Both N and N are decreases with time t because of utilization of energy. c) For a < weak comensalism of host over comensal, N falls with slower rate than N. However with increasing growth rate of host steepness increases for the N curves and falling rate is decreased comparison with N. With increasing a steepness decreases. d) For a = the steepness of N increases with a however falling of N is faster along with increasing a than N 49
VOL. 5, NO., OCTOBER ISSN 89-668 6- Asian Research Publishing Network (ARPN). All rights reserved. e) For a > strong commenslism of host over commensal it is observed that N increasing to some extent and then start falling. This tendency is observed at slow rate in case of a =. f) Because of growth rate of S, N falls much faster than N. However this fall becomes slower in case of a > Trend index N N Steepness of N a a a REFERENCES [] Bhaskara Rama Sarma. B, Pattabhi Ramacharyulu N.Ch and Lalitha. S. V. N. 9. A Recursive procedure for two species competing Eco system with decay and replenishment for one Species. Acta ciencia Indica. Xxxv M. (): 487-496. [] Cushing J.M. 977. Integro - differential equations and delay models in population Dynamics. Lect. Notes in Biomathematics. Vol., springer - reflag, Heidelbegh 9. [3] Kapur J.N. 985. Mathematical models in Biology and Medicine. Affiliated East-West 9. [4] Langlois W.E and Rivlin R.S. 957. Steady flow of slightly visco-elastic fluids. (Doctoral thesis of W- Elanglois), Brown universality. [5] Meyer W.J. 985. Concepts of Mathematical modeling. Mc-Grawhill. [6] Phanikumar N and Pattabhi Ramacharyulu N. Ch. On the stability of a commesal-host harvested species pair with limited resources. Communicated to International journal of computational cognition. [7] Srinvias N.C. 99. Some mathematical aspects of modeling in Bio-medical sciences. Ph.D. thesis, submitted to Kakatiya University. 5